Elements of Applied Bifurcation Theory pp 249-294 | Cite as

# Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems

Chapter

## Abstract

The list of possible bifurcations in multidimensional systems is not exhausted by those studied in the previous chapters. Actually, even the complete list of all generic one-parameter bifurcations is unknown. In this chapter we study several unrelated bifurcations that occur in one-parameter continuous-time dynamical systems
where

$$\dot x = f(x,\alpha ),x \in {R^n},\alpha \in {R^1},$$

(7.1)

*f*is a smooth function of (*x*,*α*). We start by considering global bifurcations of orbits that are homoclinic to nonhyperbolic equilibria. As we shall see, under certain conditions they imply the appearance of complex dynamics. We also briefly touch some other bifurcations generating “strange” behavior, including homoclinic tangency and the “blue-sky” catastrophe. Then we discuss bifurcations occurring on invariant tori. These bifurcations are responsible for such phenomena as frequency and phase locking. Finally, we give a brief introduction to the theory of bifurcations in symmetric systems, which are those systems that are invariant with respect to the representation of a certain symmetry group. After giving some general results on bifurcations in such systems, we restrict our attention to bifurcations of equilibria and cycles in the presence of the simplest symmetry group ℤ_{2}, composed of only two elements.## Keywords

Hopf Bifurcation Homoclinic Orbit Rotation Number Center Manifold Invariant Torus
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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